Optimal. Leaf size=109 \[ \frac {2 \left (5 a^2+2 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 d \sqrt {\cos (c+d x)}}-\frac {14 a b (e \cos (c+d x))^{3/2}}{15 d e}-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e} \]
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Rubi [A] time = 0.13, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2692, 2669, 2640, 2639} \[ \frac {2 \left (5 a^2+2 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 d \sqrt {\cos (c+d x)}}-\frac {14 a b (e \cos (c+d x))^{3/2}}{15 d e}-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2640
Rule 2669
Rule 2692
Rubi steps
\begin {align*} \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2 \, dx &=-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e}+\frac {2}{5} \int \sqrt {e \cos (c+d x)} \left (\frac {5 a^2}{2}+b^2+\frac {7}{2} a b \sin (c+d x)\right ) \, dx\\ &=-\frac {14 a b (e \cos (c+d x))^{3/2}}{15 d e}-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e}+\frac {1}{5} \left (5 a^2+2 b^2\right ) \int \sqrt {e \cos (c+d x)} \, dx\\ &=-\frac {14 a b (e \cos (c+d x))^{3/2}}{15 d e}-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e}+\frac {\left (\left (5 a^2+2 b^2\right ) \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 \sqrt {\cos (c+d x)}}\\ &=-\frac {14 a b (e \cos (c+d x))^{3/2}}{15 d e}+\frac {2 \left (5 a^2+2 b^2\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)}}-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 80, normalized size = 0.73 \[ \frac {\sqrt {e \cos (c+d x)} \left (6 \left (5 a^2+2 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )-2 b \cos ^{\frac {3}{2}}(c+d x) (10 a+3 b \sin (c+d x))\right )}{15 d \sqrt {\cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}\right )} \sqrt {e \cos \left (d x + c\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {e \cos \left (d x + c\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.73, size = 251, normalized size = 2.30 \[ \frac {2 e \left (-24 b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-40 a b \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24 b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+15 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{2}+6 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) b^{2}+40 a b \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-6 b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-10 a b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {e \cos \left (d x + c\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {e\,\cos \left (c+d\,x\right )}\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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